In this paper, we introduce the definition of a convex real
valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We
prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f
\geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept,
we state and prove discrete Hermite-Hadamard inequality using
the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$).
Second, we state and prove the discrete fractional Hermite-Hadamard
inequality using the basics of discrete fractional calculus.
We close the paper by defining the convexity of a real valued
function on any time scale.

We present various weighted integral inequalities for partial
derivatives acting on products and compositions of functions
which are applied to establish some new Opial-type inequalities
involving functions of several independent variables. We also
demonstrate the usefulness of our results in the field of partial
differential equations.

We study a compactness property of the operators between weighted
Lebesgue spaces that average a function over certain domains involving
a star-shaped region. The cases covered are (i) when the average is
taken over a difference of two dilations of a star-shaped region in
$\RR^N$, and (ii) when the average is taken over all dilations of
star-shaped regions in $\RR^N$. These cases include, respectively,
the average over annuli and the average over balls centered at origin.